4th International Conference on
Earthquake Geotechnical Engineering
June 25-28, 2007
Paper No. 1488
COMPARISON OF DYNAMIC PROPERTIES OF CLAYS OBTAINED
BY DIFFERENT TEST METHODS
Selim ALTUN 1, Volkan OKUR 2, A. Burak GÖKTEPE 3, Atilla ANSAL 4
ABSTRACT
For cohesive soils, an attempt for the classification of Gmax considering two relevant parameters,
namely confining pressure (σ0’) and plasticity index (PI), was carried out by hard k-means (HKM),
fuzzy c-means (FCM) and Kohonen’s self organizing map (SOM) methodologies. After determination
of descriptive statistics of the experimental data, three different methods were applied to determine the
ranges as well as representative values of Gmax using the free parameters.
The first part of the study is a description and the analysis of experimental investigations conducted to
determine the nonlinear stress-strain behaviors of clays when subjected to cyclic loadings at small
strain levels. A series of torsional shear and cyclic triaxial shear tests were performed under undrained
conditions to analyze the variations in the stress–strain properties of undisturbed saturated clay
samples consolidated to specified confining pressure. The uniform cyclic sinusoidal loading at a 0.1
Hz frequency was applied to samples with different confining pressures. Test results are presented and
evaluated in detail.
The second part of the study focuses on hard k-means (HKM), fuzzy c-means (FCM) and Kohonen’s
self organizing map (SOM) techniques to classify the complex mapping ranging from plasticity index
and confining pressure input spaces to dynamic shear modulus output space.
Consequently, comparative results were presented in terms of the methods’ clustering capabilities, and
the benefit of such an analysis in soil dynamics was evaluated. Results indicate that the dynamic
properties of clays are significantly affected by the testing technique and the initial conditions.
Keywords: Shear modulus, cyclic test, classification, fuzzy c-means, k-means, self organizing map.
INTRODUCTION
The deformational characteristics of soils, generally expressed by means of shear moduli and material
damping ratios, are crucial parameters in determining the dynamic performance of geotechnical
materials and the severity of ground shaking during earthquakes. Several researchers have attempted
to understand the nonlinear deformation characteristics of soils, as well as the variations in shear
modulus and damping ratios under cyclic loadings (Jamiolkowski et al, 1991; LoPresti et al., 1996).
Evaluation of maximum shear modulus (Gmax) is of great importance with respect to its usage in
characterizing the stiffness of a particular soil. Apart from the empirical relationships derived for the
determination of Gmax, classification of this dynamic parameter by means of relevant factors may also
1
Assistant Professor, Department of Civil Engineering, Ege University, Bornova, Izmir, Turkey
Email: selim.altun@ege.edu.tr
2
Assistant Professor, Department of Civil Engineering, Osman Gazi University, Eskisehir, Turkey.
3
Project Manager, Kolin Const Co., Akkoy Dam and HEPP, Kurtun, Gumushane, Turkey.
4
Professor, Bogazici University, Kandilli Observatory and Earthquake Research Institute, Istanbul,
Turkey.
be significant to understand the level of dynamic strength under certain conditions (Hardin and
Drenevich, 1972; Hardin, 1979).
There are more than a few laboratory based methods that can be utilized to obtain maximum shear
modulus, such as bender element, resonant column, cyclic simple shear, cyclic triaxial, and torsional
shear tests. The main discrepancies among the testing methodologies come from the fact that they
measure stress-strain variations at different strain levels and under different stress and strain boundary
conditions as well as their accuracies are different from each other. Interpretation of laboratory studies
lead to the conclusion that, soil stiffness expressed by Gmax, is strongly influenced by cyclic strain
level, void ratio, mean effective stress, plasticity index, overconsolidation ratio, and number of loading
cycles. Although an easier way to estimate Gmax is utilization of an empirical relationship, specific care
should be devoted for the errors in interpreting shear wave velocity and anisotropic properties of soils.
Although numerous empirical relationships have been formulated and seem to be consistent with
experimental observations, the measurement of dynamic soil properties (in laboratory or in-situ) is
time-consuming and requires considerable computational effort. For this reason, it is desirable to
develop a practical approach for the prediction of dynamic stress-strain parameters for soils.
Numerous attempts have been made to estimate the maximum shear modulus of a specific types of
soils using empirical expressions. However, only limited number of geotechnical engineers have been
interested in the classification of soils with respect to dynamic soil properties. (Kokusho et al., 1982;
Lanzo et al., 1997)
Studies considering the relationship between maximum shear modulus and confining pressure are also
available in the literature. Results of these studies revealed that, shear modulus increases exponentially
with the confining stresses with the exponent ranging from 0.58 to 0.01 (Saxena et al., 1988). A
number of studies were carried out for the determination and modeling of Gmax (Jovicic, 1996;
Yamashita, 2001). Attempts on understanding this parameter using fuzzy logic and neural networks
are also encountered in the literature. Akbulut et al. (2004) used neuro-fuzzy systems to model the
shear modulus and damping ratio of sand as well as rubber mixtures. Romero and Pamukcu (1997)
prepared samples having several particle size distributions, compacted to various densities, and tested
under different confining pressures to determine the effects of the parameters on sample response. Ni
et al.(1996) developed neural network model with six input variables (i.e. the standard penetration
value, void ratio, unit weight, water content, effective overburden pressure, and mean effective
confining pressure) to obtain shear modulus and damping ratio. The results revealed that the neural
network model was successful in predicting shear modulus but not in damping ratio.
Torsional and triaxial cyclic shear tests were conducted on undisturbed clayey samples under
undrained conditions for different confining pressures in this study. The tests were performed under
multi-stage cyclic loadings that were increased in each five cycles. Therefore, both initial values and
modulus reduction curves of shear modulus as well as damping ratio increases were determined by
means of cyclic torsional and triaxial shear tests.
Furthermore, an attempt to classify Gmax by means of both mean effective stress (σ0’) and plasticity
index (PI) was made using fuzzy c-means method, hard k-means algorithm, and self organizing maps
Utilizing the methods, cluster centers were obtained by σ0’ and PI parameters, and a detailed
comparison on the performances of employed methods were made. Results denoted that, all the
classification methods depicted above are capable of making the desired clustering analysis; however,
there are considerable discrepancies among the methods due to their precisions.
EXPERIMENTAL STUDY
Natural undisturbed inorganic clay samples were obtained from Kocaeli Region in the frame of a
geotechnical investigation programme after the 1999 Kocaeli earthquake. Dynamic hollow cylinder
(DHC) and dynamic triaxial test (DTT) systems were used in order to evaluate modulus and damping
characteristics of the undisturbed clayey samples.
The tests were conducted according to JGS 0541-2000, JGS 0542-2000, JGS 0543-2000, JGS 05501998 standards. Undisturbed samples were trimmed 50mm diameter and 105mm in height in DTT.
Hollow cylindrical samples were prepared by drilling and trimming with a special method. The hollow
cylindrical samples are 140 mm in height. The outer diameter and the inner diameters are 70mm and
30mm, respectively.
Both testing systems has stress controlled dynamic loading and strain controlled static loading
mechanisms. Also DTT apparatus were equipped with high sensitive displacement sensors and a load
cell in the pressure chamber to completely eliminate the effect of mechanical frictions on the measured
soil properties. For larger deformations, additional displacement gauges also introduced outside of the
chamber in order to make reliable measurements for a wide strain range. It was possible to measure
both small and large strains by using two kinds of displacement gauges in one unique test. In DHC
apparatus vertical, torsional, and lateral stresses as inner and outer cell pressures can be applied
independently to the specimen.
Undrained cyclic tests were conducted with a DTT and DHC in which cyclic loads were applied by a
servo-pneumatic actuator at a frequency of 0.1 Hz. All tests were performed on undisturbed fine
grained soil specimens under an isotropic effective confining pressure and a constant back pressure.
Samples recovered under ground water table were chosen for the tests. Eight strips of saturated
filtration sheets were introduced around the specimen to provide the faster completion of the primary
consolidation and to make accurate measurement of the excess pore water pressure during the tests.
The pore water pressure was measured at the base cap in both test apparatuses. Drainage during the
stages of consolidation was allowed through the top and base cap of the specimens. By using these
filtration sheets the 100 % completion of the primary consolidation is achieved in 3-4 hours in most of
the test samples. The samples were isotropically consolidated in the cell and with a back pressure
ranging between 200-300 kPa in order to assure the saturation. Average index properties are: natural
water content ωn=36 %, liquid limit wL= 58%, plasticity index PI=21 and the specific gravity,
Gs=2.71. The Skempton B values were between 98-100% indicating that specimens were fully
saturated.
Multi stage cyclic loading was preferred as the number of available test specimens was limited. In
multi stage loading following the consolidation of the specimen, a sequence of prescribed cyclic shear
stress was then applied starting with relatively small amplitudes with each sequence having the same
number of cycles. The amplitudes of cyclic stresses were increased in each sequence on the specimen
under undrained conditions. The number of cycles, N was chosen as the third of every 5th cycle and
analyzed to obtain a representative cyclic stress-strain behavior. Two typical test results are given in
Fig. 1.
In this manner, 52 samples having different PI values were tested using both torsional and triaxial test
apparatus in order to obtain maximum dynamic shear modulus. All the test results obtained under
different effective confining pressure are illustrated as Gmax versus PI in Fig. 2.
80
Torsional Shear
σo’=200kPa, PI=10%
70
60
G (MPa)
50
40
30
Triaxial
σo’=200kPa, PI=12%
20
10
0
0.0001
0.001
0.01
γ (%)
0.1
1
10
Figure 1. Typical cyclic torsional shear and triaxial test results
120
Torsional Shear
100
Triaxial Shear
G (MPa)
80
60
40
20
0
0
10
20
30
40
PI (%)
Figure 2. Scatter plots of maximum shear modulus vs. plasticity index for general test data
CLUSTERING TECHNIQUES
Hard k-means algorithm
Clustering (or classification) is essentially to partition the finite data into a number of clusters by
understanding the underlying structure. In order to classify the data with respect to similar features and
attributes, it is required to define a similarity measure describing the distance between pairs of feature
vectors. Therefore, a clustering technique determines optimum partitions due to a certain dissimilarity
function that measures global error extent between data points and cluster centers in feature space.
Hard k–means (HKM) algorithm, also referred to as c-means algorithm, partitions the n × m
dimensional data matrix, x, into c clusters by minimizing the dissimilarity function (J) that is defined
as (Ross, 1995):
⎡ n c
⎤
J (U*, v *) = min[J (U, v )] = min⎢
χik (dik )2 ⎥
⎢⎣ k =1 i =1
⎥⎦
∑∑
(1)
where, U is m × n dimensional partition matrix, v is c × m dimensional cluster center matrix, n is the
number of data points, c is the number of clusters, d is c × n dimensional similarity matrix, U* is
partition matrix with optimal values, v* is cluster center matrix involving optimal cluster centers, and
χ is characteristic function calculating the partition matrix (U) matrix as follows:
⎧⎪1 xk ∈ ith cluster
U = {χ1, χ2 , χ3.......χi } and χi ( xk ) = ⎨
⎪⎩0 xk ∉ ith cluster
Utilizing
Euclidean distance measure to characterize the similarity, the elements of d are calculated by:
∑ (x
m
d ik = d (xk − vi ) = xk − vi =
j =1
− vij )
2
kj
(for, i = 1 to c and k = 1 to n)
(2)
the
(3)
in which, m is the number of features, xk is kth data point and vi is the centeroid of ith cluster that can be
presented by:
vi = {vi1 , vi 2 ,......., vim }
(for, i = 1 to c)
(4)
In addition, cluster centers are calculated as follows:
n
v ij =
∑χ
ik
× x kj
k =1
(for, i = 1 to c and j = 1 to m)
n
∑
(5)
x ik
k =1
Fuzzy c-means algorithm
Fuzzy c-means (FCM) algorithm was presented by Ross (1995) as the extension of hard k-means
technique with the advantage of fuzzy set theory. In fuzzy set theory, the crisp boundary in classical
set theory that separates the inclusion and the exclusion decision is shifted with the transition region,
by means of changing the binary membership with gradual membership on the real continuous interval
[0, 1]. Thereby in fuzzy approach, uncertain belonging of a point to a set is described by partial
membership (Zahid et al., 2001).
Fuzzy c-means algorithm permits a data point’s belonging to one or more clusters utilizing
membership value concept. Therefore, elements of the partition matrix consist of membership values
varying within the interval [0, 1], and a data point can partially belong to a cluster (Ross, 1995).
Basically, fuzzy c-means algorithm calculates fuzzy partition matrix to group some of data points into
c clusters. Therefore, the aim of this algorithm is to cluster centers (centeroids) that minimize
dissimilarity function (Jm), which is given by (Lanhai, 1998):
⎡ n c
m'
2⎤
J m (U *, v *) = min [J m (U , v )] = min ⎢ ∑ ∑ (μ ik ) × (d ik ) ⎥
⎣ k =1 i =1
⎦
(7)
where, μik is the membership value of the kth data point in the ith cluster, m′ is weighting parameter
varying in the range [1, ∞], U is fuzzy partition matrix, v is cluster center matrix, and d is similarity
matrix given in Eq.3 (as in hard k-means algorithm). In addition, cluster centers are calculated using
following formulation (Ross, 1995):
n
v ij =
∑μ
k =1
m'
ik
× x kj
n
∑μ
k =1
(for, i = 1 to c and j = 1 to m)
m'
(8)
ik
where, x is fuzzy variable describing data point. In essence, fuzzy partitioning is performed through an
iterative optimization utilizing following formulation:
uik ( s + 1) =
1
2
⎡c
⎤
m ' −1
⎛
⎞
d
(
s
)
⎢ ⎜ ik
⎟ ⎥
⎢∑ ⎜ d ( s ) ⎟ ⎥
⎢⎣ j =1 ⎝ jk ⎠ ⎥⎦
(9)
Self organizing maps
The self organizing map (SOM), also referred to as Kohonen map, is an unsupervised learning
methodology reducing the dimensionality of the data utilizing self-organizing neural network
methodology (Haykin, 1996). Basically, SOMs classify input vector with respect to the grouping in
the input space. Neighbouring neurons in SOM learn to recognize neighbouring sections of the input
space; therefore, it is possible to teach both distribution and topology of input vectors. Basically, the
user-defined dimension of input vector and two-dimensional output layer forms the architecture of the
SOM. There exists a relation between the adjacent neurons, which forms the topology of the output
space. Particularly, rectangular and hexagonal lattices are common topologies in use. The algorithm is
extensively explained by Kohonen (1990).
In order to focus on the algorithm, let x be the normalized input vector randomly selected from the
input space, and wij denotes the assigned synaptic weights (connections) between input and output
layers. Output of the neuron j can be summarized as:
O j = ∑ i =1 wij X i
n
(13)
where, O, w and, X denote output vector, weight matrix, and input vector, respectively. All output
values of output neurons are set to 0; exceptionally, the winning neuron is set to 1. The node with its
weight vector closest to the input vector is considered as the winner and its weights are adjusted. In
case of a node wins the competition, the neighbors' weights are also changed, as well as further the
neighbor is far from the winner, the smaller its weight change. This process is repeated for each input
vector for a number of iterations. Obviously, different winners can be produced from different inputs.
Using the Euclidean measure of distance, the following formulation can be employed in the algorithm:
d j = X j − w ij
(14)
where, d is lateral distance vector. The most common neighborhood function is the Gaussian
neighborhood function that is used to determine the topological neighborhood as given below:
h ij = e
⎛ d ij 2
⎜−
⎜ 2σ 2
⎝
⎞
⎟
⎟
⎠
(15)
where, hij is topological neighborhood and σ is the parameter of neighborhood width. Furthermore,
weights of neighboring neurons are recalculated utilizing the following equation:
[
w ij (p + 1) = w ij (p) + η(p) h ij (p) X j (p) − w ij (p)
]
(16)
where, p is the iteration step, and η denotes learning rate parameter decreasing as the iteration
continues. As a consequence, the algorithm is terminated when the map is thought to have converged
to a stable state as soon as the assignment of the various input values to the respective winning unit
remains constant. If no noticeable changes are observed in the feature map, new input pattern is
selected and the process is restarted.
CLASSIFICATION OF THE Gmax BY CONFINING STRESS USING HKM, FCM AND SOM
The main objective of this study is to emphasize the feasibility of defining Gmax intervals for plasticity
and confining pressure values using three different clustering techniques that are fundamentally
different from the confidence interval theory. Additionally, two different dynamic testing procedures,
namely cyclic triaxial and torsional shear tests, were compared under the light of this philosophy.
The information of experimental study performed to obtain necessary data were reported elsewhere
(Altun, 2003; Okur and Ansal, 2000). Obviously, the aim is to determine whether a data pattern
belonging to the same cluster on the Gmax-PI plane has the same confining pressure. The experimental
results from 52 samples tested summarized in Figure 3. The minimum and maximum values of the
plasticity index are 9% and 35%, respectively. Accordingly, maximum and minimum values for
Gmax/σ0’ parameter are pronounced as 0.15 and 0.39. As can be derived from Figure 2 easily, Gmax
values observed at confining pressures of 100 and 300kPa are widely scattered. As a result of this,
initial application of the classification algorithms in Gmax-PI plane yielded unacceptable results.
Afterward, classification trials were made using the normalized Gmax/σ0’ values. Corresponding
normalized data points are shown in Figure 3. Accordingly, clustering in terms of normalized Gmax by
confining pressure was found to be more successful than the first trial shown in Figure 3.
Three algorithms described above were applied to classify the Gmax values in terms of the plasticity
index and the confining pressure. Solutions obtained using HKM, FCM and SOM methodologies are
summarized in Figures 3. In essence, all the methods explained above are capable of clustering the
data of the range of four particular percentages of Gmax. As mentioned before, fuzzy c-means method
utilizes the partial membership concept that quantifies the belonging of a data point to any cluster.
Uncertainty existing in experimental data can be handled using fuzzy approach better. Smoothed
membership values belonging to the three classes are given in Figure 3. It should be noted that
particular artificial membership functions for each cluster cannot be identified easily as shown in
Figure 4. In addition, clustering outcomes of FCM and HKM algorithms depend on the initial cluster
centers; therefore, the algorithms were applied many times in order to obtain the final clusters.
Additionally, the solutions of FCM methodology, which is more successful for clustering than the
other algorithms, are given in Fig. 4. Clustering outcomes obtained using three different algorithms for
triaxial and torsional shear test data are separately illustrated in Fig. 5 and Fig. 6, respectively. The
cluster centers calculated by HKM, FCM and SOM for all test data are tabulated in Table 1. In Table 2
and 3, the cluster center obtained by three clustering methodology for triaxial and torsional shear test
data are given.
It should be noted that the main objective of this study is to determine threshold values for Gmax
measurements for predefined number of classes. Because a precise estimation of Gmax value is not
possible due to soil heterogeneity and anisotropy, to characterize an interval for specific plasticity and
confining pressure measurements can be used in several generalization studies. Namely, representative
Gmax value can be predicted by plasticity data, which can be simply obtained, using such an approach.
Apart, as an alternative to confidence interval theory in statistics, different approach is presented in
this study. Consequently, the testing techniques were compared in this manner.
In essence, there are two directions of this study that can be considered as an accumulation in soil
science. First of them, an idea about self-classification of widely used dynamic test results is presented
in order to make a generalization. In detail, soil parameters are generally known to be heterogeneous,
and it is hard to determine precise values characterizing the behavior. Therefore, parameters are
usually treated with certain confidence intervals and safety factors. From this point of view, an
innovative generalization method is proposed as an alternative to traditional statistics-based
procedures. Moreover, the approach can also be utilized for conditions in which several Gmax
estimations are required by means of plasticity index and previous Gmax measurements. Therefore, a
representative Gmax value can be calculated by simply obtained plasticity index value for a certain soil
on which dynamic tests previously performed. Second consideration of this study is to compare the
results of torsional shear and cyclic triaxial tests. In order to achieve this, different tests performed on
specific clay samples are considered by classifying representative values using the clustering
techniques. Results denoted that there are worth mentioning discrepancies among the results of
clustering analyses, and the best one should be preferred carefully.
Table 1. Cluster centers obtained using the three algorithms for all test data
Cluster centers
HKM
FCM
SOM
Percentage
of Gmax (%) PI (%) Gmax/σ0’ PI (%) Gmax/σ0’ PI (%) Gmax/σ0’
11.500 0.2808
11.834
0.2757
14.195
0.2624
0-25
15.714 0.2441
18.144
0.2415
17.392
0.2489
25-50
19.154 0.2424
24.027
0.2240
22.887
0.2246
50-75
26.350 0.2093
31.133
0.1789
26.792
0.2106
75-100
Table 2. Cluster centers obtained using the three algorithms for triaxial test data
Cluster centers
HKM
FCM
SOM
Percentage
’
’
of Gmax (%) PI (%) Gmax/σ0
PI (%) Gmax/σ0
PI (%) Gmax/σ0’
0-25
25-50
50-75
75-100
12.500
18.750
24.000
33.000
0.2423
0.2300
0.2034
0.1578
12.040
18.000
23.500
32.000
0.2496
0.2198
0.2085
0.1596
14.059
17.517
23.291
27.584
0.2392
0.2303
0.2082
0.1839
Table 3. Cluster centers obtained using the three algorithms torsional test data
Cluster centers
HKM
FCM
SOM
Percentage
of Gmax (%) PI (%) Gmax/σ0’ PI (%) Gmax/σ0’ PI (%) Gmax/σ0’
0-25
25-50
50-75
75-100
11.000
17.000
21.800
27.333
0.3604
0.2826
0.2413
0.2302
27.013
10.580
16.725
21.224
0.2332
0.3639
0.2756
0.2376
14.035
17.781
22.522
25.566
0.2844
0.2565
0.2271
0.2185
0.50
0-25%
25-50%
50-75%
75-100%
Center-HKM
Center-FCM
Center-SOM
0.40
G /σo’
0.30
0.20
0.10
0.00
0
10
20
30
40
PI (%)
Figure 3. Solution by three algorithms for general data
0.50
0-25%
25-50%
0.40
50-75%
75-100%
Center-FCM
G /σo’
0.30
0.20
0.10
0.00
0
10
20
30
40
PI (%)
Figure 4. Solution by fuzzy c-means method for general test data
0.50
0-25%
25-50%
50-75%
75-100%
Center-HKM
Center-FCM
Center-SOM
0.40
G /σo’
0.30
0.20
0.10
0.00
0
10
20
30
PI (%)
Figure 5. Solution by three algorithms for triaxial test data
40
0.50
0-25%
25-50%
50-75%
75-100%
Center-HKM
Center-FCM
Center-SOM
0.40
G /σo’
0.30
0.20
0.10
0.00
0
10
20
30
40
PI (%)
Figure 6. Solution by three algorithms for torsional test data
CONCLUSIONS
In this study, three unsupervised clustering algorithms were employed to determine the thresholds of
Gmax values of fine grained soils considering plasticity index and confining pressure parameters. The
clusters were arranged by the algorithms so that the Gmax values stays under four different ranges
constituting each cluster. Following conclusions were drawn from this study:
1. It was concluded in the investigation that, the normalization of Gmax data with initial confining
pressure (σ0’) is required to achieve successful clustering outcomes.
2. FCM, HKM and SOM methodologies were utilized to cluster maximum shear modulus of clay
soils by means of plasticity index. The algorithms were gave different outcomes as a cluster
center to handle existing uncertainty in the input data.
3. Although the methodologies in this study are capable of clustering the maximum shear
modulus of soils by means of the plasticity index, analogous study should be performed on
larger databases involving different soil types. Moreover, this approach can be applied
successfully on further geotechnical engineering problems involving uncertain and ambiguous
data.
REFERENCES
Akbulut S, Hasiloglu AS, Pamukcu S. “Data generation for shear modulus and damping ratio in
reinforced sands using adaptive neuro-fuzzy inference system,” Soil Dynamics and Earthquake
Engineering, 24, Issue. 11, 805-814, 2004.
Altun S. Evaluation of cyclic behavior of soil with torsional shear test. Ph.D. Thesis. Istanbul, Istanbul
Technical University, 2003 (in Turkish).
Hardin BO and Drnevich,VP. “Shear modulus and damping in soils,” Journal of Soil Mechanics and
Foundation Division, ASCE, 98, Issue. 7, 667-692, 1972.
Hardin BO. “The nature of stress-strain behaviour of soils,” Proceedings of the earthquake engineering
and soil dynamics, ASCE, Pasadena, California, Vol. I, 1978.
Haykin S. Neural Networks, Second Edition, New Jersey: Prentice Hall, 1996.
Jamiolkowski M, Leroueil S and LoPresti DCF. Theme lecture: design parameters from theory to
practice. In: Proceedings of Geo-Coast’91. Yokohama, Japan, 1991.
JGS T 520-1990, Practice for preparing triaxial test specimens, Japanese Society of Soil Mechanics
and Foundation Engineering, 1990.
Jovicic M, Coop R and Simic M. “Objective criteria for determining Gmax from bender element tests,”
Geotechnique, 46, Issue. 2, 357–362, 1996.
Kohonen, T. “The Self-organizing map,” Proceedings of the IEEE, 78 Issue. 9, 1460-1480, 1990.
Kokusho T, Yoshida Y. and Esashi Y. “Dynamic properties of soft clay for wide strain range,” Soils
and Foundations, 22, 1-18, 1982.
Lanhai L. “Comparison of conventional and fuzzy land classification and evaluation techniques in
Oxfordshire, England,” International Agricultural Engineering Journal, 7, 1-12, 1998.
Lanzo G, Vucetic M and Doroudian M. “Reduction of shear modulus at small strains in simple shear,”
Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 123, Issue.11, 1035-1042,
1997.
LoPresti DCF, Jamiolkowski M, Pallara O, Cavallaro A and Pedron S. “Shear modulus and damping
of soils,” Geotechnique, 47, Issue.3, 603-617, 1996.
Ni SH, Juang CH and Lu PC. ‘‘Estimation of dynamic properties of sand using artificial neural
networks,” Transportation Research Record, 1526, 1-5, 1996.
Okur DV and Ansal AM. “Dynamic characteristics of clays,” Proceedings of the 3rd Japan-Turkey
Workshop on Earthquake Engineering, Istanbul, Turkey, 45-52, 21-25 February 2000.
Romero S and Pamukcu S. “Prediction of dynamic properties by low strain excitation of residual
materials,” ASTM Special Technical Publication 1275-Testing Soil Mixed with Waste or Recycled
Materials, 79-92, 1997.
Ross T. Fuzzy logic with engineering applications, New York: McGraw Hill Co., 1995.
Saxena SK, Avramidis AS and Reddy KR. “Dynamic moduli and damping ratios for cemented sands
at low strains,” Canadian Geotechnical Journal, 25, Issue.2, 353-368, 1988.
Yamashita S, Kohata Y, Kawaguchi T and Shibuya S. “International round-robin test organized by
TC-29,” Advanced laboratory stress–strain testing of geomaterials, Balkema, 2001.
Zahid N, Abouelala O, Limouri M and Essaid A. “Fuzzy clustering based on K-nearest-neighbours
rule,” Fuzzy Sets and Systems, 120, 239-247, 2001.